Linear approximation and Taylor expansion of -terms F. Olimpieri - - PowerPoint PPT Presentation

Linear approximation and Taylor expansion of λ-terms

• F. Olimpieri

Aix-Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille, France

• F. Olimpieri

Linear approximation and Taylor expansion of λ-terms 1 / 5

The pure λ-calculus

λ-terms We inductively define Λ : if x ∈ V then x ∈ Λ; If M ∈ Λ, then λx.M ∈ Λ; if M, N ∈ Λ, then MN ∈ Λ. λx.M stands for x → M. We can model functional evaluation: (λx.M)N → M[N/x]

• F. Olimpieri

Linear approximation and Taylor expansion of λ-terms 2 / 5

The pure λ-calculus

λ-terms We inductively define Λ : if x ∈ V then x ∈ Λ; If M ∈ Λ, then λx.M ∈ Λ; if M, N ∈ Λ, then MN ∈ Λ. λx.M stands for x → M. We can model functional evaluation: (λx.M)N → M[N/x]

• F. Olimpieri

Linear approximation and Taylor expansion of λ-terms 2 / 5

The pure λ-calculus

λ-terms We inductively define Λ : if x ∈ V then x ∈ Λ; If M ∈ Λ, then λx.M ∈ Λ; if M, N ∈ Λ, then MN ∈ Λ. λx.M stands for x → M. We can model functional evaluation: (λx.M)N → M[N/x]

• F. Olimpieri

Linear approximation and Taylor expansion of λ-terms 2 / 5

Linearity

Intuitive Definition A function f is linear when it uses only once its argument during the computation. Linearity for functional evaluation: The identity function is linear. Let M ∈ Λ, then (λx.x)M → M. The copy function is non-linear. Let M ∈ Λ, then (λx.xx)M → MM.

• F. Olimpieri

Linear approximation and Taylor expansion of λ-terms 3 / 5

Linearity

Intuitive Definition A function f is linear when it uses only once its argument during the computation. Linearity for functional evaluation: The identity function is linear. Let M ∈ Λ, then (λx.x)M → M. The copy function is non-linear. Let M ∈ Λ, then (λx.xx)M → MM.

• F. Olimpieri

Linear approximation and Taylor expansion of λ-terms 3 / 5

Linearity

Intuitive Definition A function f is linear when it uses only once its argument during the computation. Linearity for functional evaluation: The identity function is linear. Let M ∈ Λ, then (λx.x)M → M. The copy function is non-linear. Let M ∈ Λ, then (λx.xx)M → MM.

• F. Olimpieri

Linear approximation and Taylor expansion of λ-terms 3 / 5

Linear approximation of λ-terms

Linear logic leads to the introduction of a resource sensitive approximation of programs. Intuitively, a n-linear approximant of a term M is a version of it that uses exactly n times the argument under evaluation. We denote as T(M) the set of linear approximants of M. Lemma Let M ∈ Λ and s ∈ T(M). If s → t then there exists N ∈ Λ such that t ∈ T(N) and M → N.

• F. Olimpieri

Linear approximation and Taylor expansion of λ-terms 4 / 5

Linear approximation of λ-terms

Linear logic leads to the introduction of a resource sensitive approximation of programs. Intuitively, a n-linear approximant of a term M is a version of it that uses exactly n times the argument under evaluation. We denote as T(M) the set of linear approximants of M. Lemma Let M ∈ Λ and s ∈ T(M). If s → t then there exists N ∈ Λ such that t ∈ T(N) and M → N.

• F. Olimpieri

Linear approximation and Taylor expansion of λ-terms 4 / 5

Linear approximation of λ-terms

Linear logic leads to the introduction of a resource sensitive approximation of programs. Intuitively, a n-linear approximant of a term M is a version of it that uses exactly n times the argument under evaluation. We denote as T(M) the set of linear approximants of M. Lemma Let M ∈ Λ and s ∈ T(M). If s → t then there exists N ∈ Λ such that t ∈ T(N) and M → N.

• F. Olimpieri

Linear approximation and Taylor expansion of λ-terms 4 / 5

Some results

Theorem Let M ∈ Λ. M is computationally meaningful iff the computation for some s ∈ T(M) ends. We can define a Taylor expansion for λ-terms: Taylor formula Θ(M) =

• s∈T(M)

1 m(s)s

• F. Olimpieri

Linear approximation and Taylor expansion of λ-terms 5 / 5

Some results

Theorem Let M ∈ Λ. M is computationally meaningful iff the computation for some s ∈ T(M) ends. We can define a Taylor expansion for λ-terms: Taylor formula Θ(M) =

• s∈T(M)

1 m(s)s

• F. Olimpieri

Linear approximation and Taylor expansion of λ-terms 5 / 5